Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
From Groupprops
Statement
Suppose is a finite group,
is a subgroup, and
is a field whose characteristic does not divide the order of
(we do not require
to be a splitting field, though the splitting field case is of particular interest).
Then, the Maximum degree of irreducible representation (?) of over
is less than or equal to the product:
(maximum degree of irreducible representation of over
) times (index of
in
, i.e.,
)
Facts used
Related facts
Proof
Proof in characteristic zero
Given: is a finite group,
is a subgroup, and
is a field of characteristic zero.
has index
in
.
is the maximum of the degrees of irreducible representations of
over
.
To prove: has an irreducible representation over
whose degree is at least
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ![]() ![]() ![]() |
![]() ![]() |
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2 | Let ![]() ![]() ![]() |
![]() ![]() |
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3 | The inner product ![]() ![]() ![]() ![]() |
We need ![]() |
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4 | ![]() |
Fact (2) | |||
5 | ![]() ![]() ![]() |
Steps (3), (4) | |||
6 | The degree of ![]() ![]() ![]() |
Steps (1), (5) | |||
7 | The degree of ![]() ![]() ![]() |
![]() ![]() ![]() |
Definition of induced representation. | ||
8 | The degree of ![]() ![]() |
Steps (6), (7) |